If the order of \(\Sh (E/\mathbb {Q})[p]\)  is known to be finite and the \(p\) -adic regulator of \(E\)  is non-zero, then it is known that the order of vanishing of \(f_{E}^{alg}(T)\)  at \(T=0\)  is equal to \(g\) , the algebraic rank of \(E\) . This is a result of Schneider and Perrin-Riou (see for instance [1]}). These conditions are imposed on the elliptic curves \(E_1\)  and \(E_2\) . The truncated Euler characteristic is related to the characteristic series \(f_{E}^{alg}(T)\) . Write \(f_{E}^{alg}(T)=T^g g_{E}(T)\)  where \(g_{E}(0)\ne 0\) . The following is a direct consequence of [2]}.
